Suppose a group $G$ acts faithfully on a set of five elements, inducing two orbits of size $3$ and $2$ respectively. What group may $G$ be?
There is clearly a homomorphism $G \mapsto S_3$ and another $G \mapsto S_2$.
$|G|$ cannot be greater than $|S_2 \times S_3| = 12$, the total number of simultaneous permutations, or its action would not be faithful.
It seems to me that $D_3=S_3$, $C_6$, and $S_3 \times C_2$ are possibilities. Is this list correct and exhaustive, and how do I prove it?